Approximating entropy for a class of $\zz^2$ Markov Random Fields and pressure for a class of functions on $\zz^2$ shifts of finite type

TL;DR

This paper demonstrates that the entropy of certain $bz^2$ Markov Random Fields can be approximated exponentially fast using strip entropies, with implications for pressure computation in shifts of finite type, including models like Ising and hard core.

Contribution

It establishes exponential convergence of entropy differences for $bz^2$ MRFs and provides explicit computability of pressures for functions on shifts of finite type.

Findings

Strip entropies converge exponentially fast to the true entropy.

Pressures for certain functions are computable when their values are computable.

Results apply to models like the Ising and hard core models for specific parameters.

Abstract

For a class of $\zz^2$ Markov Random Fields (MRFs) $\mu$, we show that the sequence of successive differences of entropies of induced MRFs on strips of height $n$ converges exponentially fast (in $n$) to the entropy of $\mu$. These strip entropies can be computed explicitly when $\mu$ is a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor $\zz^2$ shift of finite type $X$. We state this result in terms of approximations to the (topological) pressures of certain functions on such an $X$, and we show that these pressures are computable if the values taken on by the functions are computable. Finally, we show that our results apply to the hard core model and Ising model for certain parameter values of the corresponding interactions, as well as to the topological entropy of certain nearest-neighbor $\zz^2$ shifts of finite type, generalizing a result in \cite{Pa}.